A nontrivial graph is \textit {$r$-connected}, for $r\ge 0$, if it has more than $r$ vertices and no two vertices are separated by fewer than $r$ other vertices. Graphs of $d$-polytopes are $d$-connected, according to Balinski (1961). The chapter also discusses a recent result of Pilaud et al (2022) on the edge connectivity of simplicial polytopes. We examine the higher connectivity of strongly connected complexes in Section 4.3. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs.Graphs of $d$-polytopes are $\floor{(d+2)/3}$-linked, but not all graphs of $d$-polytopes are $\floor{d/2}$-linked.Edge linkedness can be defined similarly by replacing vertex-disjoint paths in the definition of linkedness by edge-disjoint paths. The $d$-connectivity of a graph of a $d$-polytope is a particular case of a more general result of Athanasiadis (2009) on the connectivity of $(r,r+1)$-incidence graphs of a $d$-polytope. The chapter ends with a short discussion on the connectivity of incidence graphs.

Boolean network tomography is another well-studied branch of network tomography, which addresses the inference of binary performance indicators (e.g., normal vs. failed, or uncongested vs. congested) of internal network elements from the corresponding binary performance indicators on measurement paths. Boolean network tomography fundamentally differs from additive network tomography in that it is a Boolean linear system inversion problem in which each measurement path only provides one bit of information and hence deserves a separate discussion. This chapter introduces a series of identifiability measures (e.g., k-identifiability, maximum identifiability index) to quantify the capability of Boolean network tomography in uniquely detecting and localizing failed/congested network elements. As the definitions of these identifiability measures are combinatorial in nature and hard to verify for large networks, the discussion focuses on polynomial-time verifiable conditions and computable bounds, as well as the associated algorithms.